Optimal. Leaf size=116 \[ -\frac {29887 \sqrt {1-2 x} \sqrt {3+5 x}}{1024}-\frac {2717}{768} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {247}{480} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {3}{40} \sqrt {1-2 x} (3+5 x)^{7/2}+\frac {328757 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1024 \sqrt {10}} \]
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Rubi [A]
time = 0.02, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {81, 52, 56, 222}
\begin {gather*} \frac {328757 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1024 \sqrt {10}}-\frac {3}{40} \sqrt {1-2 x} (5 x+3)^{7/2}-\frac {247}{480} \sqrt {1-2 x} (5 x+3)^{5/2}-\frac {2717}{768} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {29887 \sqrt {1-2 x} \sqrt {5 x+3}}{1024} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 81
Rule 222
Rubi steps
\begin {align*} \int \frac {(2+3 x) (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx &=-\frac {3}{40} \sqrt {1-2 x} (3+5 x)^{7/2}+\frac {247}{80} \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {247}{480} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {3}{40} \sqrt {1-2 x} (3+5 x)^{7/2}+\frac {2717}{192} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {2717}{768} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {247}{480} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {3}{40} \sqrt {1-2 x} (3+5 x)^{7/2}+\frac {29887}{512} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {29887 \sqrt {1-2 x} \sqrt {3+5 x}}{1024}-\frac {2717}{768} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {247}{480} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {3}{40} \sqrt {1-2 x} (3+5 x)^{7/2}+\frac {328757 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{2048}\\ &=-\frac {29887 \sqrt {1-2 x} \sqrt {3+5 x}}{1024}-\frac {2717}{768} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {247}{480} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {3}{40} \sqrt {1-2 x} (3+5 x)^{7/2}+\frac {328757 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1024 \sqrt {5}}\\ &=-\frac {29887 \sqrt {1-2 x} \sqrt {3+5 x}}{1024}-\frac {2717}{768} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {247}{480} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {3}{40} \sqrt {1-2 x} (3+5 x)^{7/2}+\frac {328757 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1024 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 78, normalized size = 0.67 \begin {gather*} \frac {-10 \sqrt {1-2 x} \left (428139+1112169 x+938420 x^2+543200 x^3+144000 x^4\right )-986271 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{30720 \sqrt {3+5 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 104, normalized size = 0.90
method | result | size |
risch | \(\frac {\left (28800 x^{3}+91360 x^{2}+132868 x +142713\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{3072 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {328757 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{20480 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(103\) |
default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (-576000 x^{3} \sqrt {-10 x^{2}-x +3}-1827200 x^{2} \sqrt {-10 x^{2}-x +3}+986271 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-2657360 x \sqrt {-10 x^{2}-x +3}-2854260 \sqrt {-10 x^{2}-x +3}\right )}{61440 \sqrt {-10 x^{2}-x +3}}\) | \(104\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.61, size = 75, normalized size = 0.65 \begin {gather*} -\frac {75}{8} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {2855}{96} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {33217}{768} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {328757}{20480} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {47571}{1024} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 72, normalized size = 0.62 \begin {gather*} -\frac {1}{3072} \, {\left (28800 \, x^{3} + 91360 \, x^{2} + 132868 \, x + 142713\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {328757}{20480} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 86.25, size = 338, normalized size = 2.91 \begin {gather*} \frac {2 \sqrt {5} \left (\begin {cases} \frac {1331 \sqrt {2} \left (\frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} + \frac {3 \sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{1936} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{16}\right )}{16} & \text {for}\: \sqrt {5 x + 3} > - \frac {\sqrt {22}}{2} \wedge \sqrt {5 x + 3} < \frac {\sqrt {22}}{2} \end {cases}\right )}{25} + \frac {6 \sqrt {5} \left (\begin {cases} \frac {14641 \sqrt {2} \cdot \left (\frac {2 \sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} + \frac {7 \sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{3872} + \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{1874048} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {35 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{128}\right )}{32} & \text {for}\: \sqrt {5 x + 3} > - \frac {\sqrt {22}}{2} \wedge \sqrt {5 x + 3} < \frac {\sqrt {22}}{2} \end {cases}\right )}{25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.29, size = 63, normalized size = 0.54 \begin {gather*} -\frac {1}{30720} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (36 \, x + 71\right )} {\left (5 \, x + 3\right )} + 2717\right )} {\left (5 \, x + 3\right )} + 89661\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 986271 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (3\,x+2\right )\,{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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